As already pointed out [1]}, defining \(C(x)\)  as \(U_{L_1}\)  implies pre-measurement only with no measurement result, and leads to contradictions. An alternative choice of operation is that Charles performs a projection operation after the pre-measurement. This refines the definition of \(C(x)\)  to include both \(U_{L_1}\)  and a projection operation on \(s_1\) , i.e., \(C(x)=U^{\dag }_{L_1}(I_{F_1}\otimes |c\rangle \langle c|)U_{L_1}\) . With this refined definition, one can verify that \([A(x), C(x)]\ne 0\)  still true. Choosing \(C(x)=U^{\dag }_{L_1}(I_{F_1}\otimes |c\rangle \langle c|)U_{L_1}\)  implies operator \(C(x)\)  is from Charlie's point of view. This may not be the original intention in [2]}. However, the key point here is that either choice of \(C(x)\) , \([A(x), C(x)]\ne 0\) . The same analysis goes to operators \(B(y)\)  and \(D(y)\) , and the conclusion is that \([B(y), D(y)]\ne 0\)  for \(y\in \lbrace 1,2,3\rbrace \) .
